I can tell you the gist of a "wall crossing formula". Typically you have a space of parameters $\newcommand{\eS}{\mathscr{S}}$ $\Lambda$, a configuration space $\newcommand{\eC}{\mathscr{C}}$ $\eC$, and a parametrized moduli space $\newcommand{\eM}{\mathscr{M}}$ $\eM$ which is a subvariety $\eM\subset \eC\times \Lambda$.
We have a natural projection
$$\pi:\eM\to \Lambda. $$
The fiber of this map over $\lambda\in\Lambda$, denoted by $\eM_\lambda$, is called the moduli space corresponding to the parameter $\lambda$.
One could associate to $\eM_\lambda$ various invariants. For example, there might exists a sheaf $\eS\to\eM$ which restricts to a sheaf $\eS_\lambda\to\eM_\lambda$. We denote by $e(\lambda)$ the Euler characteristic of $H^\bullet(\eM_\lambda,\eS_\lambda)$.
It could happend that $e(\lambda)$ depends on $\lambda$, but it could depend in a rather specal way. Namely, there could exists real codimension one subvarieties $W_i\subset \Lambda$, $i\in I$, called walls, so that, if
$$ W:=\bigcup_{i\in I} W_i, $$
then $\lambda\to e(\lambda)$ is continuous on $\Lambda^\ast:=\Lambda\setminus W$. In particular, the function $\lambda\to e(\lambda)$ is constant on the connected components of $\Lambda^*$, which are called chambers.
If two chambers $C_0, C_1$ are adjacent, i.e., they sit on opposite sides of a wall $W_i$ , then $e(\lambda)$ has constant values $e(C_0)$ and $e(C_1)$ in these two chambers and a wall crossing formula will tell you what the difference $e(C_1)-e(C_0)$ is.
Here is a trivial example. Let $\Lambda=\mathbb{R}^2$. We let $(b,c)$ denote the coordinates of a point in $\Lambda$. The configuration space $\eC$ is $\mathbb{R}$ and the parametrized moduli space is
$$\eM= \bigl\lbrace (t,b, c)\in\eC \times \Lambda;\;\;t^2+bt+c=0\,\bigr\rbrace. $$
The moduli space $\eM_{b,c}$ can be identified with the set of real roots of the quadratic polynomial $t^2+bt+c$. We denote by $e(b,c)$ the number of such roots. In other words, $e(b,c)$ is the (topological) Euler characteristic of the space $\eM_{b,c}$.
In this case we have a single wall
$$ W=\bigl\lbrace (b,c)\in \Lambda;\;\; b^2- 4c=0\bigr\rbrace, $$
with two chambers,
$$ C^\pm=\bigl\lbrace \;\pm(b^2-4c)>0\;\bigr\rbrace. $$
In this case the Wall crossing formula is
$$ e(C^+)-e(C^-)= 2. $$
